The enumeration of bipartite graphs

Di!~crete M!llllenlilticr 2g I[UT~) 4:) ~7 f6) N~rth.llolhmd Puhlh lin~', ('omp~mV

THE ENUMERA'nON

O~ BJPAi FIT|g G R A P H S

Phil H A N L O N

Received 25 Mlly 1~)7N ~ev~cd 3t: M~rch H~7 blcll]i)red grnl,h~ tW lelflhg S,i ~ ~1 ~tC[ N~] il~ [illl['lI~lll ~qlllO? Y' '~VllClC ,~" (], ~. . . . tl} 7 * II~e ,~el ~ff i'rlllt~t~l~(ed I,Ic.h,red KHiph~, iliad l / t*; 11117~lOUp Ill~l¢ dl~l~rt hllH]!(,,~ Ill(, Vi~tlt'~ * ;ll~H' tc~l~th?l~slfll~ hcIwg(.tl the Itt, llet~illll~ tuitclJotl~ J'()l I~Jc~dl~ted nnd ~tllle~l,,d bh'lde ~(,~Jlttlq*h,~ z~, ~CH(*fd:Jll~! fljt~L'ijl)ll vlhl~'}~ t~ p/e{'j~gl'/ fhr* L/l:ftClilljflj~ rmirll~,n [ll~ i~ollll/'lll'dhllljlfljl~ I~j:~IIhq

|n~oduclio. T h ~ ~-IolllrJvi~] as0cct ol ~llalller~fill;; bip~rlil~: ~r ,~h~ conies in de~:rnlinin~', tile geP.l~rt~,[Jllg fuIl~Ii~);t for cl)llnected bieolored im, phs f i n n t i ~ genl~rlltinfl flJtlclJt,i1 f~r bleolored graphs, For mo~,t kinds c-f p:1aphs, prohlem~ o f this ~ort are ~olved | y applying the f o r m u l a

duc to Riddell [51. Unfortunately, this method does not work for bicohmed p~-aphs. H a r a r y and Prlns [3] nevertheless succeeded in c a r r y i n g out this ennmeral;iorl using an i n v o l v e d series o f graph t h e o r y arguments. H o w e v e r , the ~ v e m u a [ algebraic relationship they reach between the counting functions for bieoi ,.ed and connected bieolored graphs is qui~'e ..omplicatcd and fails to reflect tl ~. simple combinatorial relationship between th:~se two sets of graphs. In this paper, the standard construet!~n of graphs with n connected cornpo,lents is modified for bicolored graphs by alluwing S. x H to act on the functiol space Y× where X = { 1 , 2 , . . . , n}, T' is the ~ t af connected bicolored graphs, and /-f is the group that interchanges the vertex colors. Then DeBruijn's Generaliz~ ~en [2] is applied to show that in this case, a g~neralized form of Riddel!'s formu':; holds, thus ,giving a more direct algebraic relationship between the generating fuactions for bicolored and connected bicolored graphs. 49

~[}

pr tf¢~nlon

I. Definitions l.~el~tnilion 1. A labelled, hicolored graph B of ordt where V ( B ) = I I , 2 . . . . . n}, E ( B ) is a set of 2-e! f : V(B) ~ {J', g~ such lkat {u, ~} e E(B) =~ f(u) ~ f(~,

h a triple (V(B), E(B), [) n*. subsets c,f V(B) and

Exarnple !o 3

4

3

B:, "\',/;" I i

2

f(1)=l(2)-r,

f(3) = / ( 4 ) - [(5) = g

Let L,, denote the set of labe]!ad, bieolored graphs of order n. The group S. acts on L,, by

c~(V(BL E(B). f ) - (V(B). o'E(B), fo" ~) where erE(B)- {(cru. ~rV}:~u, v}c-E(B)}. The critical observation in showing that L. i:~ closed under this action is that

{¢ru, o-v}e¢yE(B)~{u, vl, aE(B)::~ f ( u ) ~ f ( v ) ~

fcy I(gu)7/fo" t(o-v).

The orbits of trds action are called b:colored graphs and the set of all bicolored gra[~h~; ~f order ,r i~ denoted ~,,. It is worth r.othlg that the two colors r and g ha~:e distinct identities. In particular, r :, r r

g g

g

g

are unequal as bicolored graphs. Let H he the group H = {(r)(g), (rg)}. H acts on ~,, as follows: Let I1 ~ H, let e ~ . and let (V(B), E(B),f) be a representati~¢e element from 0 (recall d is an orbit hence a set). T h e n define IIO to be: the orbit of ~ . containing

(V(B) E(B), fir). It is easy to show that this action is well-defined (independent of tl~e choice of

(V(P), E ( B ) , f) from O). A bicolored ~;raph O E ~,, which is fixed under the action of H is called symmetric. Example 2. Let

g

~

g

g

Then ;) is a symmetric, bieolored graph.

"t'lwem~metari,m id h p m ,' ~era;,i~,;

51

[)e|Jni|ton 2, Lc| crY.S,, and (~ ~-r , ir i'~ ~al]¢d ~!rl al~to~lorp/li.~m nf (; if trE!G)= E ( G ) , The automorphisms iff G a), ~imply the autnml~rphism~, ,d' ~11c I:'bclled graph ( V ( G L E(G)), ()h~er,,e Ihat an autonl¢lrpl;isnl of G need nc!l he. a color-preservin~.r autornorphism ,ff th,' hieolored graph G, !n particular, let H be t'le labelied bicolored Rraph

o=f ,/',, \/_ I

5

2

3

7

6

4

8

fl(l)-/(2)-t(3)-fGf)-g,

f(5)

/((.t

/(7)'f(~,)

r

(, - (1, 5)(2.~';)(3, 6)(4.7) 's an automorphism of B but not a color-preservinfz automnrl)hism of B since [~r ~J: f. Fh,' group of automorphisrns of G is denmed /'(G). The follow:ng technical lemma wili be of use in the next section. I' characterizes 5 vnmetric bicolored ,~raphs as those having an automorphisrr which changes the c:)lor of every vertex. Lemma 1. 0 is :G,mmetric iff V G - ( V ( G I , f r ~(ut¢ f(u) for all u,: V(G).

U(GI, t')eO. "~(rEI'(G) sl,~h that

F'roof. 0 is symmetric iff HO = 0 iff g G ~ 0, ~]~r ~ S. with

(V(G), E(G). Irg)flt - ( V ( G ; , rrE(G).fo- ~) iff V G e O , 3 c r e - / ' ( G ) f,,' '(u)Vu~ V(G).

with (rg)f=f¢r ~ iff V G ~ , 9

3(re:l'(G)

with f ( u ) ¢

Definition 3. A labelled, bicolored graph ( V ( G ) , E ( G ) , fl) is comwcted if the underlying labelled graph (V(G), E ( G ) ) is eomlected. Define F(x, y) and G(x, y) to bc the generating functions

t:(x. y~ = ~ (f;,..x" +f,,,,,y"). .

G(x,y)=

i

~ (g,,,.x"+R.,,,y')

where f..(~=number of nensymmetric bicolored graphs pRn, half the nu nber of !:ymmetric bicolored graphs, f~,.. half the number of symmetric bicolored graphs. g , , . o - n u m h : : of connected nonsymmetric bicolored graphs plus half the number of connected symmetric bicolored graphs, go.,, = half the .-mnbcr of connected, symmetric bicolored graphs.

52

P Hanlon

2. £a~e~blafing F ( x v) and G ( x , y) Following the notation of Harary and Palmer [3], let Z(S,,~ x S,,) be the cycle h~dex o" the action of S,,~ xS,~ on the set MxN={i,i):ie[I.2

.....

m}.i~[1.2 .....

tt is easy to v~rify that

xC(~q . . . . . h~. k~ . . . . . k.) where :he sum:; are takcD over all vectors of non-negati'.e integers with ~" ih~ = m and }] ik, = n. where A(hl .....

h.,)

111", i~"h~! '

B(k, .....

k,,) -~?

where n! l ik,kl] ~

and where C(k ......

hm, k . . . . . . k,,)= izI [z] S[~3I r

I,

,

I

the notation (i, j) meaning greatest common divisor of i and j and the notation [i, j] meaning the least common multiple of i and j. For e~ ample. Z ( S 2 x S~) = i'~(s'? + 3 ~ s ~ + 2 s ~ + ~ + 3s~+ 2 s J , , , + ~ 2-~SlS~+ ~ 4 s 2~ + 2 s ~ + 2 S s ) . =l~(Sl

Let N x K r be the set N x I ~ = { i , i ) : i e [ 1 , 2 . . . . . n},~e{1,2. . . . . . fi}} and let JS,,]s, denote tl:e group acting on N x N which is generated by S. xS,, :ogether with o- = t 1 , 1 ) ( 2 , 2.). •. (n. ii). The exponentiation group [S.] s~ has order 2(n!) 2 and has (S,, ×S, ) as a subgroup of index 2. Let z~,, denole the cycle index of the subset [S,,]s~-S,, x S,,; to be precise let 2,, = z([S.l~.)-~z(s,

xs.).

It is not hard to show that -

1

Z.=--~ ~] A ( h ~ . . . . . n. ~,, J,,0

h,,)D(h~ . . . . . h,,)~2(h~ . . . . .

h.,)

where as before, the sum is taken over all n-tuplcs of non-negative integers satisfying Y i& =, n. where A ( h , . . . . . h.) = ~,[, II

n! ia, h, ! '

The enumeratlot~ of

srtitef,raoM

53

where

D(hl,.

•.,

h~)= 1~I

s, %

i o~td

and where

E(h~,

•. •

, !- '2)+r:,21;, i l)h h , h.)= ~ s2, I ' l s2Ia}

The notation It~'.'.] meaning the gre ~test integer in I/2. _1 2 H-F o r example, Z2-:dsls-z ~4). For a thorough discussion of the~e t,~o ~ruups and thcir cycle indices see Harary and Paimec [3, Chapter 4]. It is worth noting that both Z(S,, × S...) and Z, are easily computable.

3. Entmmerafit~g

Mparfile

~aphs

The foilowing theo, .~m gives, for bico!orable graphs, the ;,,~propriate analoguc of RiddeIl's formula

G(x):

~ z(s,,)[s, --, c:(~')]

The notation s~--> 2 means that e,ch , should be replaced by the Similarly, ~ --~ G(x ~, 0) indicates th" t ea 'h s~ shou!d be re01aced by the function G ( # , 0). Observe that F(z, y) can be ~ompated using part theorem; once F(x, y) is ~nowr, G(x, y) can be obtained recursiwJy (B?.

Thed~rem

(B)

number 2. generating (A) of the using part

1.

F(x, y) = ~. (½(Z(S,,)[s~ --~ G(x'. 0)] .

1

.kZ(S.)[s2~--)(;(y2~,O).s.,.t

~C(0, y 2'~) )).

Proof. First (A) will be verified• By Polya's Theorem, Z(S.,, ×S,,)[s,- 2] is the number of orbits of [unctions mapping X = M × N to Y={0, 1} (see [~],. If ~ is nuch a function, map the orbit containing ~ onto the bicolcred graph co*~tainin Z tbe abelled bicolored graph B~ = (V(B~), E(B~), f 0 where s ( p ~ ) = ((i, m + i}: g(i, i) = 1}

54

p.

Hanlon

and where f~(]) . . . . . f ~ ( m ) - r e d and f~(m+ 1) . . . . . f~(m+n):-green. It is easy to s l o w fnat this map is well-defined (independent of the choice of ~,) and that the map sends the set of orbits contained in Y~ 1-1 and onto the set of bicolored grapks with m red points and n green points. T o compietc the proof of (A), it remains to sho~ "hat the number of ,,s~,mmetric bicolored graphs with n vertices of each color 2- ts~--~ 2]. Make the same a~sociation as e~bove between the orbits of Y~ u m he acticn of [S,~]s~ and the set of bicolored grapLs with n points of each eolo, By Lemma l. symmetric bicolored graphs are exac, those which corre;pond to orl:its containing functions which are stabilized by a t.ermuta'don in the cose~ o'(S,, xS~), where o is as d e f n e d in the miedle of page a. If h is such a fun':tion ar.d .f

F(h)~{ac-[5~]S':hu

~=h},

then

(F(h)NIS,,xS,,))~_F(h)

and

IFlh):(F(h)n(S,,xS,,)~l,- 2.

So, each function stabilized by a permutation in the coset o-(S,, × S,) is counted exactly once in 2;,[s~ ~ 2], which complete~ the proof of (A). To prove (B), recall file action of the group H ={Ir)(b), (rb)} defined on page 2; a bicolored graph was deSned to be symmetric if it is fixed under this action by the whole group .H (equilalently by (rb)). So a simple application of Burnside's Lemma gives that F(x, y) enumerates the set of orbits under this action. Let X - { l , 2 . . . . k}, I;~t Y be the set of connected bicolored g~aphs and let ~u,, be the set of bicolored graphs with precisely k connected components. Let Sk act on X, and let i:/ act ot Y and on ~o~k~. Each function f in Y" corresponds in a natural way to an orbit o~ ~,k~ namely that orbit containing the bicolored graph with ~onnected components {[(1), f(2) . . . . . f(k)}. It is straight-forward to show that this correspcndence extends to a I - 1 , onto correspondence between the orbits of Y* (acted on t,y Hs~I and the orbits of ,°)o~ So,

~'~'(x, y )=

~

w(f).

f.v~v' Del3ruijn's generalization of Polya's Theorem giw:s

f, v~y, where

d=y wr~

The enumeration # ,ipartite~raphs

5~

and where the last sum is taken over all d-, c e s in the disjoint cycle decomposition of h. We have two choices for h; if h - ( , ~ ( b ) tN:n Y breaks into a product of 1-cycles so

w,(Y, (r)(g)) = G(x', O) for all i. If h = f i g ) then the non-symmetric, connected, bicolored graphs fall into 2-cycles, whereas the symmet-ic, connected bicoiored graphs fall into 1-cycles. Thus

w2~(Y: (rg)) - G(y ~', 0), w2,.,(Y: (rl;)) = G(0, yc2~-.). Putting this together with the observation that every bicolored graph must have some number k/> 1 of connected components gives

F(x.

y)= ~ (~'lz(&)[x,- .

O(x', o)] k I +Z(Sk)[x2, ~ G(y 2~, 0), x2;~, ~ G(O, ~,2, .,~][).

Definition 4. A labelled graph G - (V(G), E(G)) is called bipartite if 3 U. V~_ V(G) with U ¢ 0 , V ¢ 0 and U - V ( G ) - V , such that {r,, v 2 } ~ ; V ( G ) : f f v ~ e U and v ~ V (or v~ ~ V and t,2 E UL An unlabelled graph (i;omc,rphism class of labelled graphs) is bipartite if each labelled graph contained withir~ it is bipartite. The same application of Burn.,ide's Lemma given on ,page 7 gives us that G(x, y) enumerates the set of e-',its of connected bicolored graphs under the action of H - { ( r ) ( b ) , (rb)}. Since lJ aPows the colors to be interchanged, G(x, y) enumerates the set of connected, bipartite graphs (observe that if G is a connectec bipartite graph then the decomposition V ( G ) - U U V is uniqeeiy defined). Corollary 1. Let B(x ) - ~ _ ~ b.x" and C(x) = Y,~ ~ c.x '~ be defined by b,,(c,,) is the number o[ bipartite (connected bipartite) graphs with n ~ertices. Then

C(x)-'~G(x, x) and B(x)=expQ~,~)-

..

Lastly we consider how the second equation given in Theorem I car, be inverted to express G(x, y) in terms of F(x, y). The key observation ; that both F(x, y) and G(x, y) are a sum of two generating functicns in one varial ;e; neither one has terms of tile form x~y ~ for i,]>~ 1. In particular we t'ave

F(x, y) ~ F(x, 0 ) + F(0, y)

56

p, Hanlon

and so F(x, O) and F(O, y) a" ~ given by the first , ouatio::: in T h e o r e m 1. A l s o

G(x, y) = G(x. 0)-, G(0. y) and so to find G(£, y) it s u i t e s to find G(x,O) at, Letting y = 0 in T h e o r e m i we have

. ( 0 , y).

'~+ F(x, O) = 2 ~ Z(S.)[x, ~ G ( x ' , 0)] aad so ~(n)

+

G(x,O)= Y . - - I o g ( 1 ,, = i

~t

2U(x",0)).

T h e latter equation follows by a well-known reversion formula due to C a d o g a n Letting x = 0 in T h e o r e m 1 yields + F ( 0 y) = ~ ~ Z ( & )[s2~ --> G ( y 2~, 0), sz,,, ~ G(O, y2,+,j]. ,, n and ~o

[~, G(y~',O)\

1 + 2 / ' ( 0 , y ) = exp t , ~ , ~ ) e x p

ft 6, G(O, y~'+')~ ~ , ~ ),

Hence

)2 ~(°"/'+ ) og(l+2v(o y~) ~20{y:',o~ ,,]

~TT

'

,.,

2

Thus G ( 0 , y) = ~',~7/*(2k ( i o g +( l +l)2 F { 0 , 2 / , + l

y 2 + ))

,-,,L~G

. . ~. . . . . . , 0").

q'able 1 was compw:ed from these formulas using m o d e s t a m o u n t s (3 minates) of c o m p u t e r time: it cain easily be extended beyond p - 14. In this table c. = # of p

% I

2 3 4 5 6 7 8 9 I(I 11 12 13 14

hp 1

l i ~ i I ]' 4~ 182 73{! 4032 25598 212780 224174:<10 ~ 3 110337xl0 ~

1

2 3 7 13 35 88 303 I I 19 5479 323(13 251135 2.52772x |0 ~ 3.389591 x lfP

The enumeration ,,( ~ ~a~ite graphs

connected bipartite graphs witr. p ve~-t~ es, b ~ = # vertices.

57

of bipartite graphs with p

4. Conchlsion Lines can be included as an enumeration p a r a l a e t e r by leplacing s, ~ 2 wi~h s~---~ l + w ' in Theorem I(A) and tber. making the ,ppropriate ci'anges in Theorem I(B) and Corollary 1. The t e d niques u~;ed here can be extcnded to count connected k-colored grapl)s for all k, but unlortunately this sheds no light on the general problem of counting k-co;orable g:aphs. Unlike the bicolorabk, case, connected ~-colorable graphs are not nezessarlly uniquely k-colorable fo: ~>3.

References [ I] CC, Cadogan, The M6bius function and connected graphs, J Cem',Glatorial Theory ii B (1971) 193-200. [~.] N.G, DeBrui!'l. Polya's theory of counting, in: E,F. geckenhach ed., Applied Combmalodal Mathematics ~gdilexuNew York, 196~,) ]44-lad [3~ F Harary and E. Palmer, Graphical Enumeration (Academic Press. New York, 1973). i4] F. Harary and G. Prlns, Enumer:'inn of bicolorahle graph., Canad J. Math.. 15 (1963~ 23?-248 [:~] RJ. RidJt'II, Contributions to lhe I i )ry of condensation, Di:,;ertation, Univ of Michigan, Ann Arbor, 1951.